A155828 -id:A155828 - cp's OEIS frontend

A252911 Irregular triangular array read by rows: T(n,k) is the number of elements in the multiplicative group of integers modulo n that have order k, n>=1, 1<=k<=A002322(n).

1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 0, 0, 2, 1, 3, 1, 1, 2, 0, 0, 2, 1, 1, 0, 2, 1, 1, 0, 0, 4, 0, 0, 0, 0, 4, 1, 3, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 1, 2, 0, 0, 2, 1, 3, 0, 4, 1, 3, 0, 4, 1, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 3, 0, 4

Offset: 1

Geoffrey Critzer, Dec 24 2014

Row sums are A000010.

Column 2 = A155828(n) = A060594(n) - 1.

			1;
1;
1, 1;
1, 1;
1, 1, 0, 2;
1, 1;
1, 1, 2, 0, 0, 2;
1, 3;
1, 1, 2, 0, 0, 2;
1, 1, 0, 2;
1, 1, 0, 0, 4, 0, 0, 0, 0, 4;
1, 3;
1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4;
1, 1, 2, 0, 0, 2;
1, 3, 0, 4;
T(15,2)=3 because the elements 4, 11, and 14 have order 2 in the modulo multiplication group (Z/15Z)*. We observe that 4^2, 11^2, and 14^2 are congruent to 1 mod 15.

Alois P. Heinz, Rows n = 1..250, flattened
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.

Cf. A000010, A002322, A054522, A060594, A155828.

with(numtheory):
T:= n-> `if`(n=1, 1, (p-> seq(coeff(p, x, j), j=1..degree(p)))(
         add(`if`(igcd(n, i)>1, 0, x^order(i, n)), i=1..n-1))):
seq(T(n), n=1..30);  # Alois P. Heinz, Dec 30 2014

Table[Table[
   Count[Table[
     MultiplicativeOrder[a, n], {a,
      Select[Range[n], GCD[#, n] == 1 &]}], k], {k, 1,
    CarmichaelLambda[n]}], {n, 1, 20}] // Grid

A265120 Irregular array read by rows: Row n gives the number of elements in the multiplicative group mod n, (Z/nZ, *), that have order d for each divisor d of the exponent of the group.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 1, 1, 2, 2, 1, 3, 4, 1, 3, 4, 1, 1, 2, 4, 8, 1, 1, 2, 2, 1, 1, 2, 2, 6, 6, 1, 3, 4, 1, 3, 2, 6, 1, 1, 4, 4, 1, 1, 10, 10, 1, 7, 1, 1, 2, 4, 4, 8

Offset: 2

Geoffrey Critzer, Dec 01 2015

The exponent of the multiplicative group mod n is Carmichael lambda(n) given in A002322.
The row lengths are tau(lambda(n)) = A000005(A002322(n)) = A066800(n).
The invariant factor decomposition of (Z/nZ,*) is given in A258446.
The row sums are phi(n) = A000010(n).
It appears that column 2 is A155828.

			{1}
{1, 1}
{1, 1}
{1, 1, 2}
{1, 1}
{1, 1, 2, 2}
{1, 3}
{1, 1, 2, 2}
{1, 1, 2}
{1, 1, 4, 4}
{1, 3}
{1, 1, 2, 2, 2, 4}
{1, 1, 2, 2}
{1, 3, 4}
{1, 3, 4}
{1, 1, 2, 4, 8}
{1, 1, 2, 2}
{1, 1, 2, 2, 6, 6}
{1, 3, 4}
{1, 3, 2, 6}
{1, 1, 4, 4}
{1, 1, 10, 10}
{1, 7},
{1, 1, 2, 4, 4, 8}
The row for n=21 reads: 1,3,2,6 because the multiplicative group mod 21,  (Z/21*Z,*) is isomorphic to C_6 X C_2. The exponent of this group is 6. This group contains one element of order 1, three elements of order 2, two elements of order 3, and six elements of order 6.

Cf. A000005, A000010, A002322, A066800, A155828, A258446.

f[{p_, e_}] := {FactorInteger[p - 1][[All, 1]]^
    FactorInteger[p - 1][[All, 2]],
   FactorInteger[p^(e - 1)][[All, 1]]^
    FactorInteger[p^(e - 1)][[All, 2]]};
fun[lst_] :=
Module[{int, num, res},
  int = Sort /@ GatherBy[Join @@ (FactorInteger /@ lst), First];
  num = Times @@ Power @@@ (Last@# & /@ int);
  res = Flatten[Map[Power @@ # &, Most /@ int, {2}]];
  {num, res}]
rec[lt_] :=
First@NestWhile[{Append[#[[1]], fun[#[[2]]][[1]]],
     fun[#[[2]]][[2]]} &, {{}, lt}, Length[#[[2]]] > 0 &];
t[list_] :=
Table[Count[Map[PermutationOrder, GroupElements[AbelianGroup[list]]],
    d], {d, Divisors[First[list]]}];
Map[t, Table[
   If[! IntegerQ[n/8],
    DeleteCases[rec[Flatten[Map[f, FactorInteger[n]]]], 1],
    DeleteCases[
     rec[Join[{2, 2^(FactorInteger[n][[1, 2]] - 2)},
       Flatten[Map[f, Drop[FactorInteger[n], 1]]]]], 1]], {n, 2,
    25}] /. {} -> {1}]

cp's OEIS Frontend

A252911 Irregular triangular array read by rows: T(n,k) is the number of elements in the multiplicative group of integers modulo n that have order k, n>=1, 1<=k<=A002322(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica